DERIVATIVE OF a TO THE POWER x

Derivative of a^x with respect to x.

Let y = a^x.

In y = a^x, we have variable x in exponent.

y = a^x

Take logarithm on both sides.

ln(y) = ln(a^x)

Apply the power rule of logarithm on the right side.

ln(y) = xln(a)

Find the derivative with respect to x.

(Since a is a constant, ln(a) is also a constant. When we find derivative xln(a), we keep the the constant ln(a) as it is and find the derivative of x with respect to x, that is 1)

Multiply both sides by y.

Substitute y = a^x.

Therefore, the derivative a^x is a^xln(a).

Solved Problems

Find ᵈʸ⁄dₓ.in each of the following.

Example 1 :

y = e^x

(x and y are variables and e is a constant)

Solution :

In y = e^x, we have constant e in base and variable x in exponent.

y = e^x

Take logarithm on both sides.

ln(y) = ln(e^x)

Apply the power rule of logarithm on the right side.

ln(y) = xln(e)

(The base of a natural logarithm is e, lne is a natural logarithm and its base is e)

ln(y) = xln_ee

In a logarithm, if the base and argument are same, its value is 1. In ln_ee, the base and argument are same, so its value is 1.

ln(y) = x(1)

ln(y) = x

Find the derivative with respect to x.

Multiply both sides by y.

Substitute y = e^x.

Example 2 :

y = 7^2x

Solution :

In y = 7^2x, we have variable in exponent.

y = 7^2x

Take logarithm on both sides.

ln(y) = ln(7^2x)

Apply the power rule of logarithm on the right side.

ln(y) = 2xln(7)

Find the derivative with respect to x.

Multiply both sides by y.

Substitute y = 7^2x.

Example 3 :

y = 3^√x

Solution :

In y = 3^√x, we have variable in exponent.

y = 3^√x

Take logarithm on both sides.

ln(y) = ln(3^√x)

Apply the power rule of logarithm on the right side.

ln(y) = √xln(3)

Find the derivative with respect to x.

Multiply both sides by y.

Substitute y = 3^√x.

Example 4 :

y = 2^ln(x)

Solution :

In y = 2^ln(x), we have variable in exponent.

y = 2^ln(x)

Take logarithm on both sides.

ln(y) = ln(2^ln(x))

Apply the power rule of logarithm on the right side.

ln(y) = ln(x) ⋅ ln(2)

Find the derivative with respect to x.

Multiply both sides by y.

Substitute y = 2^ln(x).

Example 5 :

y = 5^sinx

Solution :

In y = 5^sinx, we have variable in exponent.

y = 5^sinx

Take logarithm on both sides.

ln(y) = ln(5^sinx)

Apply the power rule of logarithm on the right side.

ln(y) = sinx ⋅ ln(5)

Find the derivative with respect to x.

Multiply both sides by y.

Substitute y = 5^sinx.

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